arXiv subjects
Publications and source records attributed to Matthias Ludewig.
We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincaré-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.
We discuss the solution theory of operators of the form $\nabla_X + A$, acting on smooth sections of a vector bundle with connection $\nabla$ over a manifold $M$, where $X$ is a vector field having a critical point with positive linearization at some point $p \in M$. As an operator on a suitable space of smooth sections $Γ^\infty(U, \V)$, it fulfills a Fredholm alternative, and the same is true for the adjoint operator. Furthermore, we show that the solutions depend smoothly on the data $\nabla$, $X$ and $A$.